What is the integral of ce^{-cx} ?

The integral of ce^{-cx} is -e^{-cx}+C

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integral of ce^{-cx}

\int c e^{- c x} d x

- e^{- c x} + C

Solution steps

\int c e^{- c x} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= c \cdot \int e^{- c x} d x

Apply u-substitution

: \int - \frac{e ^{u}}{c} d u

= c \cdot \int - \frac{e ^{u}}{c} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= c (- \frac{1}{c} \cdot \int e^{u} d u)

Use the common integral:

\int e^{u} d u = e^{u}

= c (- \frac{1}{c} e^{u})

Substitute back

u = - c x

= c (- \frac{1}{c} e^{- c x})

Simplify

c (- \frac{1}{c} e^{- c x}) : - e^{- c x}

= - e^{- c x}

Add a constant to the solution

= - e^{- c x} + C

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